New results

Explicit formulas

Other amusing examples

Without further assumptions on C or J, there is no hope at present for bounding N from above, so the best we can do is seek examples with large N.

The problem naturally splits into two parts, according as J
is simple or split. Here an abelian variety is said to be
“simple” if and only if it is not isogenous to a product
of abelian varieties of lower dimension, and “(completely) split”
if it is isogenous to a product of elliptic curves;
in our case J is two-dimensional, so it must be one or the other.
Note that the elliptic curves are allowed to be defined over some
extension of **Q**; thus we use “simple” to mean
“absolutely simple”: not split over any number field.

In the non-simple (split) case, the state of the art
is represented by the paper “Large torsion subgroups
of split Jacobians of curves of genus two or three”
by Everett Howe, Franck Leprévost, and Bjorn Poonen
(*Forum Math.***12** (2000) #3, 315-364).
There one finds examples of curves or families of (C,J,p,N)
with J split for various values of N, the largest of which is
63.
The constructions hinge on the torsion structure of the elliptic curves
whose product is isogenous with J.
These are controlled by various “modular curves”, whose study
is an active and highly developed topic in number theory.

In the simple case, the theory of modular curves is no longer available.
Torsion points of high order are usually constructed by forcing C
to have two or three non-Weierstrass points that support several
divisors linearly equivalent to zero. This method is described
in pages 84-87 of J.W.S.Cassels and E.V.Flynn's book
*Prolegomena to a middlebrow arithmetic of curves of genus 2*
(LMS Lecture Notes **230**), Cambridge University Press,
1996. This method was used by Leprévost to construct rational
N-torsion points for N as high as 29
(in his paper “Jacobiennes décomposables de certaines courbes
de genre 2: torsion et simplicité”,
*J. Théorie des Nombres de Bordeaux* **7**
(1995) #1, 283-306) and 30
(in an unpublished 4/1996 preprint, reported by Everett Howe).
While Leprévost found only a single curve
with a 29-torsion divisor, he obtained an infinite
(one-parameter) family of curves with N=30.

Bjorn Poonen has a program that computes, for most genus-2 curves C, all points that differ from their hyperelliptic conjugates by torsion divisors; his paper that develops the requisite theory and describes the program is available online in pdf or compressed dvi. He confirmed that for the N=39 curve these four pairs of points, together with the six Weierstrass points, are the only points of C, even over the complex numbers, that differ from their hyperelliptic conjugates by torsion divisors.

Poonen also observed that J, while simple, appears to have
real multiplication by **Q**(sqrt(2)), which would make
J conjecturally modular; he reports using Qing Liu's program
to determine the arithmetic conductor of J: it is
245^{2}=7^{4}5^{2},
so J should be isogenous to a simple factor
of the Jacobian J_{0}(245) of the modular curve
X_{0}(245)=X_{0}(7^{2}5),
and not to a factor of J_{0}(M) for any M<245.
Again as with N=39, good reduction at 2 can be confirmed
by “uncompleting the square” mod 4 to obtain an equivalent formula
for the curve, here

There are over a hundred examples over **Q**
with five point pairs (both with and without a rational
Weierstrass point), and many nonconstant families with
four point pairs parametrized by **P**^{1}.
The above six-point curve was found by specializing
the (q, 5q, 13q, 29q) family

has four points
(x,y)=(inf,inf^{3}), (0,1-t), (-1,1), and (1,-1)
over **C**(t) giving q,2q,8q,13q,
and the specialization t=5 has two further points,
with x=-2 and x=-130/77, that yield divisors that differ from 4q and 38q
by a 2-torsion point. A more complicated family of curves
with a rational Weierstrass point as well as 6q,8q,9q,13q
specializes to

The points on C(b) giving q,5q,13q,29q are
(x,y)=(inf,inf^{3}), (0,-(b+1)), (1,1), and (-1,2b+1).
The choice of coefficients 1,5,13,29 may seem bizarre, but in fact
{-29,-13,-5,-1,1,5,13,29} is the lexicogrphically first symmetric set
of 2*4 odd integers satisfying the necessary condition that all nonzero
sums of elements of the set be distinct. Likewise
{-13,-8,-2,-1,1,2,8,13} is one of the first such sets
if we drop the parity condition.

Michael Stoll also computed the canonical heights of q on C(b),
and of its specialization to b=-5/2, finding 1/357 and .000797...
As far as we know, .000797... is the smallest positive canonical height
yet exhibited for a point on the Jacobian of a curve of genus 2
over **Q**. Over **P**^{1},
the record is 1/546, belonging to the q,2q,8q,13q curve over
**Q**(t) exhibited above.

In the more familiar case of genus 1 (elliptic curves), there are questions of similar flavor concerning points of low positive height; one can also consider integral points on the minimal (Néron) model of an elliptic curve as an analogue of rational points on a curve of genus 2. Some remarkable examples of nontorsion points with low height and/or many integral multiples are listed here.